Questions: Simplify the following expression. Assume that each variable is positive. ∛(108 y^8)

Solution Steps

First, we factor the radicand \(108 y^8\) into a product of perfect cubes and other factors. We can write: \[ 108 = 2^2 \cdot 3^3 \] Thus, the expression becomes: \[ \sqrt[3]{108 y^8} = \sqrt[3]{2^2 \cdot 3^3 \cdot y^8} \]

Next, we separate the perfect cubes from the other factors: \[ \sqrt[3]{2^2 \cdot 3^3 \cdot y^8} = \sqrt[3]{3^3} \cdot \sqrt[3]{2^2} \cdot \sqrt[3]{y^8} \]

We simplify the cube roots of the perfect cubes and the remaining factors: \[ \sqrt[3]{3^3} = 3 \] \[ \sqrt[3]{2^2} = 2^{2/3} \] \[ \sqrt[3]{y^8} = y^{8/3} \]

Combining all the simplified terms, we get: \[ 3 \cdot 2^{2/3} \cdot y^{8/3} \]

Final Answer